Thank you very much, tomh, for your take on what the white powder under the vinyl tiles could be. I had no clue what this substance was, and I felt concerned because there is so much of the powder. I will definitely try what you suggested of putting water w/ vinegar on the powder.
I was able to locate the report of the analysis (for asbestos) of the samples from the basement floor. The test of a small sample of the white powder found under the vinyl tiles revealed the following results:
-Asbestos disp stain: Negative
-Texture: Nonfibrous, loose, powdery, gritty.
-Specific method: Area % by visual examination.
-Description--ABS: No asbestos seen. 1% cellulose.
So the above findings support what you said about the white powder being salt or lime that has leached from the cement. I've just never seen anything like this white powder before, and in some areas there is a very heavy concentration of the powder, like maybe 1/4" deep of powder under some tiles.
If you have any more comments, please leave them because I'll check back here. Thanks!!